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The theoretical underpinning of molecular structure studies has largely been based on the fixed nuclei approximation, which adiabatically separates the electronic and nuclear dynamics under the assumption that the nuclei are nearly fixed with respect to electron motion and the electrons follow the nuclear motion instantaneously. The assumption led to the so-called Born–Oppenheimer approximation, and the notion of stationary states that instantaneously adjust themselves to any nuclear configurations in space. The success of the Born-Oppenheimer approximation is attributed to the large difference between nuclear and electronic masses. The approximation has been an important tool in molecular structure studies, and even to this day, nearly all quantum chemical computations of molecular wavefunctions make use of the approximation. In the cases in which the Born-Oppenheimer approximation breaks down, it is used as a point of departure in quantum chemical computations. however, the non-adiabatic interactions arising from the breakdown of the Born–Oppenheimer approximation are of crucial importance in chemical reactions because nearly every chemical or biological process involves non-adiabatic events. The difficulty in describing the nuclei quantum mechanically arises from the fact that the nuclei are much more massive than the electrons, engendering the wavelengths of the quantum matter waves for nuclei much shorter than those of electrons. The nuclear component of the wave function is spatially much more localized than the electronic component. For many decades, the theoretical underpinning of non-adiabatic molecular structure and the difficulty in quantum mechanical treatment of nuclear dynamics remained one of the most difficult challenges in theoretical chemistry. Beginning in the 1980's, theorists in Japan [1–9] began to question the validity or the accuracy of the Born–Oppenheimer approximation for bound states and they reformulated the hartree-Fock and CI theory with intrinsic non-adiabaticity. In his pioneering work on non-Born-Oppenheimer electronic structure, Fukutome [1] put forth a non-adiabatic method in solving the Schrӧdinger equation of molecular systems with strong or intrinsic non-adiabaticity. The method is based on the fact that the electrons undergo orbital motion with nuclear coordinate dependence and the nuclei exchanging recoil momenta. In a series of seminal studies on nuclear quantum effects, Tachikawa et al. [3,5,8], Shigeta et al. [4,6], and Nakai et al. [7,9] formulated novel non-Born-Oppenheimer electronic structure methods, the so-called non-adiabatic molecular orbital (MO), to describe the nuclear motion. Both electronic and nuclear wavefunctions are simultaneously solved with the fully variational Hartree-Fock method, by which exponents and centers of Gaussian-type function (GTF) basis sets are optimized as well as the linear combination of GTF coefficients [3]. All the light nuclei and electrons are treated quantum mechanically to include nuclear quantum effects, such as zero point energy, isotope effects, and nuclear tunneling. The initial studies employed mean-field approaches that resulted in somewhat overlocalized nuclear densities due to the lack of sufficient electron-nuclear dynamic correlation. Subsequent studies [4,6–9] focused on the development of correlated non-adiabatic MO approaches. The present JCCJ special issue on Quantum hydrogen edited by Professor M. Tachikawa addresses the hydrogen nuclear quantum effects in a series of articles [10–22] with a focus on

Special Contribution to The Quantum Hydrogen Issue of <b><i>Journal of Computer Chemistry, Japan</i></b>